Problem: Simplify; express your answer in exponential form. Assume $t\neq 0, q\neq 0$. $\dfrac{{t^{-3}q^{-5}}}{{(t^{5}q^{-5})^{-4}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${t^{-3}q^{-5} = t^{-3}q^{-5}}$ On the left, we have ${t^{-3}}$ to the exponent ${1}$ . Now ${-3 \times 1 = -3}$ , so ${t^{-3} = t^{-3}}$ Apply the ideas above to simplify the equation. $\dfrac{{t^{-3}q^{-5}}}{{(t^{5}q^{-5})^{-4}}} = \dfrac{{t^{-3}q^{-5}}}{{t^{-20}q^{20}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{-3}q^{-5}}}{{t^{-20}q^{20}}} = \dfrac{{t^{-3}}}{{t^{-20}}} \cdot \dfrac{{q^{-5}}}{{q^{20}}} = t^{{-3} - {(-20)}} \cdot q^{{-5} - {20}} = t^{17}q^{-25}$